1. Let n denote an estimate calculated from a sample of size n. What does it mean to say n is unbiased? What does it mean to say n is consistent?

2. Suppose you have n observations on the random variables y, x, and z. You perform the following procedure: (a) Regress y on x obtain the estimated coefficient, , and the predicted values, y = x . (b) Regress y on x and z. Prove that: (a) The estimated coefficient on x in the second regression is equal to . (b) The estimated coefficient on z in the second regression is equal to zero. (c) The R2 from the second regression is equal to 1.0.

3. Suppose you have n observations on the random variables y, x, and z. You perform the following procedure: (a) Regress y on x obtain the estimated coefficient, , and the residuals, u = y x . (b) Regress u on x and z. Prove that the estimated coefficient on z in the second regression is numerically equal to the coefficient on z you would obtain if you regress y on x and z.

4. Suppose X is a random variable that follows the Bernoulli distribution with parameter p: each draw of X in a sample of n observations, X1, ..., Xn is equal to 1.0 with probability p and equal to zero with probability (1 p). Derive the method of moments estimator of p, denoted p.hat

5.You issue the Stata commands:

1. drop _all

2. set obs 1000

3.gen u = rnormal(5,1)

4. gen x = rnormal() .

5. gen y = 10 + x + u

6. reg y x

7. predict thisvar, resid

8. predict thatvar

9. summ thisvar thatvar

10 reg thisvar thatvar

(a) In response to the command in line 9, what would expect Stata to report the means of thisvar and thatvar to equal (either on average, or exactly: state which holds)?

(b) What will the regression coefficient on thatvar equal in line 10?